We want to have a first understanding of the effect of the coupling
between inflaton and radiation perturbations and how a sound speed
\(c_s <1\)  might change the results for the power spectrum in DBIWI. To
possibly do something analytical and to obtain a feeling of these
effects, we can start by considering some simplifying assumptions.
Since we are interested in the strong dissipation regime \(Q>1\) , we can
start by neglecting the contribution of quantum fluctuations to the
power spectrum. For \(Q > 1\) , the contribution from the thermal
stochastic fluctuations \(\xi _\Upsilon \)  in Eq. (REF )
dominates over that from the quantum fluctuations \(\xi _q\) . Moreover,
we can also analytically solve Eq. (REF ) explicitly by
neglecting the slow-roll order corrections (and likewise the metric
contributions, which also give slow-order corrections only). Under
these simplifying assumptions, the authors in
Ref. [1]} have explicitly shown that the power
spectrum grows with \(Q_*\)  like \(\Delta _{\mathcal {R}} \propto \Delta _{{\cal R},c=0} Q_*^{3c}\) , with
\(\Delta _{{\cal R},c=0} \simeq \frac{\sqrt{3}}{4\pi ^{\frac{3}{2}}}\frac{H^{3}_{*}T_{*}}{\dot{\phi }^{2}_{*}}Q_{*}^{\frac{1}{2}},\) 
